Did I Deduce Problem 4-25 in Spivak's Book Correctly?

[kakarotyjn]

Let c be a singular k-cube and p:[0,1]^k \to [0,1]^k a 1-1 function such that p([0,1]^k ) = [0,1]^k<br /> <br /> and \operatorname{det} p&#039;(x) \ge 0 for x \in [0,1]^k.If \omega is a k-form,show that <br /> <br /> \int\limits_c \omega = \int\limits_{c \circ p} \omega

Note that
\int\limits_c \omega = \int\limits_{[0,1]^k } {c*\omega } = \int\limits_{[0,1]^k } {(f \circ c)(\det c&#039;)dx^1<br /> <br /> \wedge ... \wedge dx^k }
\int\limits_{c \circ p} \omega = \int\limits_{[0,1]^k } {(c \circ p)*\omega } = \int\limits_{[0,1]^k } {(f \circ c<br /> <br /> \circ p)(\det (c \circ p)&#039;)dx^1 \wedge ... \wedge dx^k } = \int\limits_{[0,1]^k } {(f \circ c \circ p)((\det c&#039;) \cdot<br /> <br /> (\det p&#039;))dx^1 \wedge ... \wedge dx^k }

did I deduce it right?If it's right,how to prove

 

[fresh_42]

Looks good, but you have to apply the transformation theorem on $p$ as the final step.